Engineering Analysis/Linear Independence and Basis

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Linear Independance[edit]

A set of vectors are said to be linearly dependant on one another if any vector v from the set can be constructed from a linear combination of the other vectors in the set. Given the following linear equation:

The set of vectors V is linearly independent only if all the a coefficients are zero. If we combine the v vectors together into a single row vector:

And we combine all the a coefficients into a single column vector:

We have the following linear equation:

We can show that this equation can only be satisifed for , the matrix must be invertable:

Remember that for the matrix to be invertable, the determinate must be non-zero.

Non-Square Matrix V[edit]

If the matrix is not square, then the determinate can not be taken, and therefore the matrix is not invertable. To solve this problem, we can premultiply by the transpose matrix:

And then the square matrix must be invertable:

Rank[edit]

The rank of a matrix is the largest number of linearly independent rows or columns in the matrix.

To determine the Rank, typically the matrix is reduced to row-echelon form. From the reduced form, the number of non-zero rows, or the number of non-zero columns (whichever is smaller) is the rank of the matrix.

If we multiply two matrices A and B, and the result is C:

Then the rank of C is the minimum value between the ranks A and B:

Span[edit]

A Span of a set of vectors V is the set of all vectors that can be created by a linear combination of the vectors.

Basis[edit]

A basis is a set of linearly-independent vectors that span the entire vector space.

Basis Expansion[edit]

If we have a vector , and V has basis vectors , by definition, we can write y in terms of a linear combination of the basis vectors:

or

If is invertable, the answer is apparent, but if is not invertable, then we can perform the following technique:

And we call the quantity the left-pseudoinverse of .

Change of Basis[edit]

Frequently, it is useful to change the basis vectors to a different set of vectors that span the set, but have different properties. If we have a space V, with basis vectors and a vector in V called x, we can use the new basis vectors to represent x:

or,

If V is invertable, then the solution to this problem is simple.

Grahm-Schmidt Orthogonalization[edit]

If we have a set of basis vectors that are not orthogonal, we can use a process known as orthogonalization to produce a new set of basis vectors for the same space that are orthogonal:

Given:
Find the new basis
Such that

We can define the vectors as follows:

Notice that the vectors produced by this technique are orthogonal to each other, but they are not necessarily orthonormal. To make the w vectors orthonormal, you must divide each one by its norm:

Reciprocal Basis[edit]

A Reciprocal basis is a special type of basis that is related to the original basis. The reciprocal basis can be defined as: